Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Approximation of the Mulliken charges and dipole moments
of the oxygen atoms of aluminophosphate sieves
A.V. Larin a , D.P. Vercauteren b,∗
a
Department of Chemistry, Moscow State University, Leninskie Gory, B-234, Moscow 119899, Russia
b Laboratoire de Physico-Chimie Informatique, Facultés Universitaires Notre Dame de la Paix,
Institute for Studies in Interface Sciences, Rue de Bruxelles 61, B-5000, Namur, Belgium
Received 22 September 1999; received in revised form 2 February 2000; accepted 16 February 2000
Abstract
Distributed multipole analysis on the basis of periodic Hartree–Fock (PHF)-type calculations, using the CRYSTAL95
code is applied to 12 aluminophosphate molecular sieves. Several approximations of the dependence of the Mulliken atomic
charges and dipole moments of the crystallographically independent oxygen atoms calculated with the STO-3G, ps-21G∗ ,
and 6-21G∗ basis sets are obtained with respect to three oxygen parameters (i.e. average value and difference between the
Al–O and P–O bond distances, and Al–O–P angle). Some deflections from these approximate forms for the different oxygen
types are discussed. © 2001 Elsevier Science B.V. All rights reserved.
Keywords: Mulliken charges; Dipole moments; Multipole analysis; Aluminophosphate
1. Introduction
Cationic forms of aluminophosphates are an important class of thermoresistant molecular sieves and
are perspective candidates for numerous chemical
applications. Recent studies with ab initio molecular
dynamics confirmed the importance of the framework
geometry for the activation of methanol [1]. Simultaneous analyses of the geometry for a wide series of
molecular sieves should be performed together with
the consideration of the resulting electrostatic field, as
its influence on the decrease of the activation barrier
of acetylene acylation has been recently presented [2].
Interestingly, these “geometry-field” relations provide the necessary data for less computing intensive
methods like quantum mechanical/molecular mechanical methods [3], which require the knowledge of
∗
Corresponding author. Fax: +32-81-72-45-30.
the electrostatic field and a correct estimation of the
long-range interactions within molecular sieves such
as zeolite frameworks. The electrostatic interactions
clearly dominate in numerous adsorption processes
within most of the aluminosilicates as it has been
confirmed by both spectroscopic studies of adsorbed
species [4] and theoretical estimations [5]. Close
spatial structures between aluminophosphates and
aluminosilicates a priori allow the importance of the
electrostatic interactions within the aluminophosphate
frameworks to be ascertained.
Usually the electrostatic field is only qualitatively
simulated by the Mulliken charges [6,7]. However, the
importance of the consideration of higher-order multipoles for a correct calculation of the interaction energy
between CO adsorbed over a TiO2 slab has been recently shown [8]. Point charges can be fitted if the field
value is already known, for example, from periodic
Hartree–Fock (PHF)-type calculations with the CRYS-
1381-1169/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved.
PII: S 1 3 8 1 - 1 1 6 9 ( 0 0 ) 0 0 4 5 5 - 6
74
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
TAL code [9]. Unfortunately, the latter approach can
be hardly applied to estimate the field within frameworks with a large number of atoms per unit cell (UC)
with arbitrary Si/Al or Al/P ratio also including compensatory ions, i.e. most of the zeolites which really
present an interest within the most current industrial
applications. Then alternative approaches to estimate
the electrostatic field should be proposed.
In our previous studies [10,11], we used an approximate scheme, developed by Saunders et al.
[6], of distributed multipole analysis related to the
atomic positions within 14 all-siliceous analogues of
aluminosilicate zeolites. The scheme [6] has been implemented in the CRYSTAL95 ab initio PHF LCAO
code for periodic systems [9]. As it was shown [6],
distributed multipole analysis allows to obtain electrostatic field values with a precision below 1% using
multipole moments up to the fourth order. In our
work, a common type of approximate dependence for
the Mulliken charges (i.e. the moments of zeroth order within the scope of the scheme [6]) with different
basis sets (STO-3G and 6-21G in [10], and ps-21G∗
in [11]) was proposed for each crystallographically
independent type of oxygen and silicon atoms. More
precisely, two-dimensional analytical functions with
respect to the average Si–O distance and Si–O–Si angle [10] for O atoms and one-dimensional analytical
functions with respect to the average Si–O distance
[11] for Si atoms were derived. A comparison of the Si
and O charges estimated with these analytical dependences confirmed their good agreement with results
of large scale PHF-type calculations for all-siliceous
mordenite [7] and silicalite [12]. Provided that such
approximations in terms of the internal coordinates of
each sieve atom could be obtained for higher multipole
moments, electrostatic field values could be calculated
precisely within any arbitrary all-siliceous zeolite.
Our approximate scheme is also a priori very appropriate for the case of aluminophosphates because the
smaller P–O bond distances would lead to a sharper
increase of the bielectronic integrals as compared to
aluminosilicate frameworks [10,11]. Hence, the direct
calculation with the PHF CRYSTAL95 code and a
higher quality basis set even for an aluminophosphate
framework with a moderate number of atoms per UC
could be more expensive.
In this work, estimations of the dependences of the
oxygen atomic charge and dipole moment are pre-
sented for 12 aluminophosphate (ALPO) structures
with ratio Al/P = 1. This set of models was chosen
because it does not require the consideration of any
cation and it allows to apply the CRYSTAL95 code
within a reasonable limit of computing time. The dependences for the O multipole moments together with
analogous approximate dependences for the multipole
moments of the Al and P atoms [13] could be the
key to estimate the electrostatic field created by the
atoms of larger frameworks whose direct solution is
beyond the capabilities of the computing platforms and
electronic structure codes currently available. Similar
types of expressions for the multipole moments up to
octupole for Al, P, and O atoms of the ALPO sieves are
presently under study and will be presented in a future
paper. The next step would then be to evaluate the field
value for sieve contents of Al/P > 1 including cations.
In Section 2 of this paper, we discuss briefly the
basis sets used together with the characteristics of
the considered ALPO frameworks. In Section 3, we
develop the approximate functions of the atomic O
charges with respect to the internal parameters. In Section 4, approximations of such type of dependence of
the O dipole moments are presented. The use of the
two lowest multipole moments (i.e. charge and dipole)
for the framework atoms leads to a better estimation
of the electric field with respect to the one obtained
by considering the Mulliken charges only as it will be
demonstrated in the last section.
2. Theoretical aspects
The theoretical bases for the solution of the
Hartree–Fock equation in three dimensions considering periodic boundary conditions have already widely
been described in the literature [9,14,15]. The aluminophosphate structures were chosen on the basis
of a relatively small number of atoms per elementary
unit cell (UC). The characteristics of the 12 ALPO
frameworks taken from the MSI database [16,17] are
given in Table 1.
The minimal STO-3G basis set [20] was applied to
all 12 considered systems. With the Durant–Barthelat
pseudo-potential ps-21G∗ basis set for Al and P and
6-21G∗ for O [9] (named hereafter basis ps-21G∗ ), and
with the 6-21G∗ basis on all atoms, the SCF scheme
converged properly for the AST, ATN, ATO, and CHA
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
75
Table 1
Symbol, number of atoms per unit cell (UC), of different Al, P (nP = nAl ) and O types, of atomic orbitals (AO) per UC, and symmetry
group of the aluminophosphate sieves [16,17], all of them corresponding to Al/P = 1
Name
Symbola
Atoms/UC
nAl /nO
AO/UC (STO-3G)
Symmetry group
MAPO-39
AlPO4 -16
AlPO4 -31
AlPO4 -34b
AlPO4 -11
AlPO4 -41
AlPO4 -18
AlPO4 -5
AlPO4 -12-TAMU
AlPO4 -D
AlPO4 -C
AlPO4 -8
ATN
AST
ATO
CHA
AEL
AFO
AEI
AFI
ATT
APD
APC
AET
24
30
36
36
60
60
72
72
72
96
96
108
1/4
2/3
1/4
1/4
3/11
4/13
3/12
1/4
3/12
4/16
2/8
5/19
152
190
228
228
380
380
456
456
456
608
608
684
I4
F23
R3
R3
Ibm2
Cmc21
C2/c
P6cc
P21 21 2
Pca21
Pbca
Cmc21
a
b
[18].
[19].
sieves only. In the case of the ps-21G∗ basis, we optimised the exponents of the 3sp0 orbital of P and Al as
0.20 and 0.07 a.u.−2 , respectively, and of the 2sp0 orbital of O as 0.28 (instead of 0.13, 0.12 and 0.37 a.u.−2 ,
respectively, in [21]) for the AST structure. Too diffuse
functions on Al led, however, to a slower SCF convergence, i.e. up to 20–25 iterations. We also optimised
the exponents of the d polarisation functions as 0.5
and 0.25 a.u.−2 for the P and Al atoms, respectively,
and as 0.8 for oxygen. With the 6-21G∗ basis, the upper mentioned low 3sp0 exponent of Al appeared to be
not appropriate. The sp0 and d exponents for Al, P, and
O were again optimised as 0.14, 0.15, 0.45 and 0.35,
0.50, 0.72 a.u.−2 , respectively, in the case of AST.
All computations with the CRYSTAL95 code were
realised on an IBM 15-node (120 MHz) scalable
POWER parallel platform (with 1 Gb of memory/CPU). For all cases, the thresholds for the calculations were fixed to 10−5 for the overlap Coulomb,
the penetration Coulomb, and overlap exchange, to
10−6 and 10−11 for the pseudo-overlap exchange, and
to 10−5 for the pseudo-potential series for all levels
of basis sets. The computations with the ps-21G∗ and
6-21G∗ basis were executed directly without keeping
the bielectronic integrals files. The whole SCF convergence (9–10 cycles) requires around 50 and 70 min
for AST (the same for ATN) within the ps-21G∗ and
6-21G∗ basis sets, respectively, and nearly three times
more for the CHA and ATO sieves. The calculation of
the respective moments takes usually below 6–7 min.
3. Approximation of the Mulliken atomic charges
of the framework oxygens
Using the CRYSTAL95 code, 110 crystallographically independent atomic oxygen charge values were
calculated with the minimal STO-3G basis set for
all 12 studied aluminophosphate forms (ALPOs)
(Table 1) and 15 different oxygen charge values for
AST, ATN, ATO, and CHA sieves with the ps-21G∗
or 6-21G∗ bases. A representation of the O atomic
charges as a function of the T1 –O–T2 angle (ϑ) and
average T–O distance R = R(ROT1 + ROT2 )/2 (in Å)
within several all-siliceous zeolite forms:
Q00 (R, ϑ) = a1 R n + a2 (R − R0 )m cos(ϑ − ϑ0 )
(1)
Ti corresponding to Si atoms, or herein to Al, P atoms,
was already discussed [10,11]. The same representation is sufficient to reach a root mean square deviation
(RMSD) of 2.73% between the STO-3G calculated
values and the approximated ones within the ALPOs.
The parameters of function (1) are given in Table 2.
The differences between the calculated and the approximated Mulliken charges are presented in Fig. 1a.
Despite the particular important difference between
the Al–O and P–O bond lengths is disregarded with
function (1), the proposed approach leads to a rather
good approximation as compared to a total charge
variation of ±14% (Table 3). A better fit is achieved
when all three coordinates characterising the internal
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A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Table 2
Parameters of the approximate functions (Eqs. (1) and (2)) evaluated from the calculation, with STO-3G, ps-21G∗ , and 6-21G∗ basis sets,
of N crystallographically independent oxygen charge values of the ALPO sieves
Equation
Basis set
N
a1
n
a2 × 10
1
2
2
2
STO-3G
STO-3G
ps-21G∗
6-21G∗
106
110
15
15
0.522
2.414
0.711
0.042
0.356
−0.905
−0.032
1.295
−0.152
−0.678
−0.152
−0.012
R0 × 102 (Å)
113.1
−0.651
−0.652
−63.39
m
ϑ 0 × 102 (rad)
RMSD (%)
−2.900
1.157
0.453
0.410
−1.408
−0.150
−0.325
−2.693
2.73
1.55
0.43
1.45
Fig. 1. Differences between the calculated Mulliken oxygen charges and the approximated ones (in |e− |) obtained via (a) Eq. (1) and (b)
Eq. (2) with respect to the internal coordinates, i.e. average distance R (in Å) and Al–O–P angle (in rad) of all ALPO frameworks.
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Table 3
Variations of the oxygen atomic charge values of the ALPO sieves
calculated with STO-3G, ps-21G∗ , and 6-21G∗ basis sets
Basis set
N
Q0min
Q0max
Range (%)
RMSD (%)
STO-3G
ps-21G∗
6-21G∗
110
15
15
−0.8162
−0.9106
−0.4107
−0.6157
−0.8635
−0.3661
±14.0
±2.8
±6.5
1.55
0.43
1.45
geometry of each O atom with respect to its neighbours are used. For convenience, we thus replaced the
two different Al–O and P–O bond lengths by the average bond distance R upper mentioned and by the
bond anisotropy 1R = ROAl − ROP (in Å). A second modification of formula (1) was introduced allowing for a slightly better precision by considering
an exponential function with respect to the average
bond distance instead of a power presentation [11].
Finally, a six-parameter exponential form with a coupling term between the bond anisotropy 1R and the
angle Al–O–P was tested to fit the O charges in the
considered ALPOs:
Q00 (R, 1R, ϑ) = a1 enR + a2 em(1R−R0 ) cos(ϑ−ϑ0 )
(2)
Eq. (2) allows to reach a better RMSD = 1.55% for
the approximation of the Mulliken charges calculated
with the STO-3G basis set (Table 2).
An interesting point is to analyse the variations of
the three different contributions to the atomic charge
obtained with the proposed function (2). The contribution of the “coupling” term between the angle Al–O–P
and bond anisotropy 1R, i.e. the last term in Eq. (2), is
around 25% of the total charge value with the STO-3G
and ps-21G∗ basis sets. The smaller values of the m
parameter (Table 2) confirm a softer behaviour of function (2) with respect to the 1R coordinate with a higher
quality basis set. Hence, the exaggerated dependence
on the internal coordinates obtained with the minimal STO-3G basis is useful to find a main functional
form whose parameters would vary with the basis set
applied.
The structure of the AlPO4 -34 sieve (CHA
with Al/P = 1) optimised with a plane wave approach [19] was considered here together with other
three-dimensional structures, obtained via X-ray
diffraction. The respective charges and absolute val-
77
ues of the dipole moment (see Section 4) of the
framework oxygens are satisfactorily fitted and do
not demonstrate any particular behaviour. This justifies the choice of the ALPO structures studied herein,
i.e. without any preliminary optimisation in order to
find the geometry dependences. Earlier, it was shown
that the O charges calculated with CRYSTAL for
the optimised all-siliceous CHA zeolite [22] are in
agreement with values coming from an analogous
charge dependence approximation fitted using a set
of non-optimised structures [10].
The differences between the charges calculated
with CRYSTAL and the approximated ones using
function (2) for the 110 crystallographically independent oxygen atoms within the studied ALPOs are
presented in Fig. 1b. Evidently, the points fitted with
(2) are located within a narrower slab close to zero
as compared to those given in Fig. 1a. The distribution of the RMSD values with respect to the bond
anisotropy 1R does not reveal any correlation, which
confirms the satisfactory choice of the dependence
with respect to 1R in Eq. (2). An average charge
variation through all the ALPOs (Table 3) can be
evaluated from the comparison between a simple
one-dimensional linear fit (dotted line) and expression
(2) (solid line) in Fig. 2 in the case of the STO-3G
basis. The variations range to ±14 (Fig. 2a) and to
±2.8 and ±6.5% (Fig. 2b) for the STO-3G, ps-21G∗ ,
and 6-21G∗ bases, respectively, which is appreciable
as compared to the respective RMSD values of 1.55,
0.43 and 1.45% obtained with function (2). A strong
variation of the absolute charge values between the
two split-valence basis sets is a consequence of a
strong shift in the polarisation exponent for the Al
atom. As the value 0.07 a.u.−2 was not appropriate
for the 6-21G∗ basis, a higher optimised value led to
a serious change of the charges. In order to illustrate
it, both split-valence basis sets with the same polarisation exponents were considered for the CHA case
(as accepted for 6-21G∗ , see Section 2). The O charge
values of −0.4504, −0.4361, −0.4562, −0.4376|e− |
obtained with ps-21G∗ are essentially closer to those
calculated with the 6-21G∗ , i.e. −0.3883, −0.3740,
−0.3941, −0.3759|e− |, as compared to the charge
values computed with a smaller Al exponent (ps-21G∗
case in Table 3). However, we considered the ps-21G∗
basis set for all four structures (AST, ATN, ATO,
and CHA) with this small exponent value on Al,
78
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Fig. 2. Calculated Mulliken oxygen charges (in |e− |) (circles) and approximated ones via linear one-dimensional function (dotted line) and
Eq. (2) (solid line) with respect to the average distance R (in Å) for all ALPO frameworks: (a) STO-3G; (b) 6-21G∗ .
because the respective total energy for the CHA sieve
was strongly more stable: −1847.3416 instead of
−1846.6406 a.u. The lower absolute charge values are
in agreement with the idea of more covalent bonds
in the ALPO frameworks than in the all-siliceous
systems [23].
One could suggest that the RMSD value of the
charge approximation via function (2) is close to a
lower precision limit considering that several characteristics of the respective Al and P neighbour atoms
are not taken into account. Such characteristics are, for
example, the average T–O distances which influence
the atomic Mulliken charges [13] in the same manner
as for the Si atoms [11], or the point symmetry group
of the TO4 tetrahedra. This limitation within approximation (2) could be indirectly proven via a closer analysis of the charges of the various O atoms presenting
similar internal parameters and different deviation values from those obtained with function (2). This can
be seen by analysing the AEI framework (Table 4),
whose atomic charges are usually nicely approximated
with Eq. (2). First, only three oxygen atoms among the
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
79
Table 4
Characteristics of several O, Al, and P atoms of the AEI and AST frameworks and deviations (charge error) between the Mulliken charges
obtained with the STO-3G basis and the ones obtained using the approximate function (2) (respective parameters are given in Table 2)
Sieve
Oxygen type
R (Å)
1R (Å)
ϑ (◦ )
Neighbour atom types
AlO4 /PO4 symmetries
Charge error (%)
AEIa
O(6)
O(10)
O(3)
O(4)
1.6229
1.6229
1.6095
1.6390
0.2110
0.2097
0.2110
0.2097
147.4
146.7
150.5
143.5
Al(1)/P(3)
Al(2)/P(1)
Al(2)/P(2)
Al(2)/P(1)
C1 /C1
C2v /C1
C2v /C1
C2v /C2v
2.34
−0.97
−0.94
1.67
ASTb
O(2)
O(3)
O(1)
1.5595
1.5618
1.6119
0.2960
0.3117
0.3563
180.0
180.0
146.4
Al(1)/P(2)
Al(2)/P(1)
Al(1)/P(1)
C3v /Td
Td /C3v
C3v /C3v
1.00
0.94
2.50
a
b
Numbering of the atoms is from [16,17].
The same from [24].
12 different crystallographically independent types in
AEI present charge deviations higher than the calculated RMSD = 1.55% (one of the oxygens is not presented in Table 4). Secondly, atoms O(6) and O(10)
have very similar structural characteristics (Table 4).
The estimation error using approximate form (2) is
larger for O(6) linking two neighbour AlO4 and PO4
tetrahedra with a similar C1 point symmetry group, as
compared to O(10), for which the closest AlO4 and
PO4 have different point symmetries. Thirdly, comparing O(3) and O(4) presenting different spatial characteristics, we observe a higher error for O(4) positioned between AlO4 and PO4 with a similar C2v
symmetry than for O(3) whose neighbour tetrahedra
present C2v and C1 groups. Other examples of the relation between the symmetry and error value is given
for the AST form (Table 4). A smaller error is indeed
sought for the O(2) and O(3) charges connecting two
TO4 tetrahedra of different symmetries C3v and Td
than for O(1) linking tetrahedra of the same symmetry
(C3v ).
These observations of the influence of the TO4 symmetry, completed below by one example in Section
4, require further studies, because they cannot be directly confirmed for all other frameworks, like AET
and ATT, whose charges approximated by expression
(2) are relatively poorer. No TO4 tetrahedra within
ATT present a C1 symmetry, while all tetrahedra of
AET are very close or correspond exactly to the C2v
symmetry. Considering the most precise O position in
terms of symmetry coordinates relative to each type of
tetrahedron surely would serve to find a more precise
dependence.
4. Approximation of the dipole moments of the
framework oxygens
The importance of the consideration of the field
contribution from the atomic dipoles was shown in
reference [8], wherein the electrostatic field above a
TiO2 surface was simulated with different approximation levels. We illustrate this by comparison of electrostatic potential (EP) values obtained with 6-21G∗
basis (Fig. 3a) relative to the EP representation at three
levels based on the neglection of parts of the atomic
multipole moments for the CHA framework. Generally, the inclusion of the moments up to fourth order
should be sufficient for a correct calculation of the
potential [6]. The three levels of the EP evaluation
include Mulliken charges (L = 0 in Fig. 3b) on all
atoms, charges and dipoles (L = 0 and 1 in Fig. 3c)
on all atoms, and all atomic moments up to third order
(L = 3 in Fig. 3d). The EP map was calculated in the
Al–O(1)–P (angle 155.16◦ ) plane of the CHA framework (Fig. 3a) applying the POTM option available in
CRYSTAL95 [9]. In order to show the EP behaviour
within an area available for a small adsorbed molecule,
the EP was calculated within a space expanded by 4 Å
(7.56 a.u.) from the Al–O side and from the right of
the O–P side. The three highest peaks at the upper left
corner in Fig. 3a correspond to the Al, O(1), and P
atomic positions; the additional fourth peak of lower
EP value comes from an oxygen located below the Al
outside of the plane. As soon as the EP value of any
atom displaced from the plane is lower than the one
of the Al, O(1), and P atoms, the atom can be looked
at a better resolution. In order to visualise the allowed
80
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Fig. 3. (a) Electrostatic potential values EP(L) (a.u.) and EP differences presented as (1 − EP(L)/EP(6)) × 100 (%) obtained with (b)
Mulliken charges (L = 0) on all atoms, (c) charges and dipoles (L = 0 and 1) on all atoms, and (d) all atomic moments up to third order
(L = 3) relative to the potential representation based on the sixth-order moments (L = 6). The moments are calculated with the 6-21G∗
basis sets with respect to the Al–O(1)–P plane of the CHA framework [19]. L value corresponds to the upper atomic moment considered
for the EP calculation.
space within the plane, the all EP scale was limited by
the range from 0 to 3 a.u. only. From this view, one
immediately notices that a small adsorbed molecule
can be located in the right upper corner corresponding
to a plane passing between two adjacent four rings of
the CHA.
Comparing Fig. 3b–d, one can see that the EP differences (1−EP(L)/EP(6)) are large along the area near
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
81
Fig. 3. (Continued).
the zero EP value (see Fig. 3a) with all three levels of
computation (b–d). This cannot, however, be considered as a drawback of the three representations because
even a negligible shift of the zero EP value should lead
to a sharp EP difference. Then, on one hand, one can
notice large differences between the L = 0 and L = 0
and 1 evaluations of the EP map in the above men-
tioned available part of internal space. If the charges
only lead to EP differences up to 100% (b), the EP
difference in case (c) does not exceed 20%. The EP
map corresponding to the moments up to second order is not presented as it is only slightly better than
the L = 0 and 1 EP map. On the other hand, a sharp
improvement of the map quality is observed while
82
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
adding octupole moments as in the (d) case, which
could be explained by the relative higher contributions
of the third-order moments of the Al and P atoms.
For their configurations which are close to tetrahedral ones, permanent non-zero electrostatic moment
values lower than octupoles are forbidden. However,
both T atoms have small dipoles and quadrupoles due
to distortion of the AlO4 and PO4 tetrahedra. Speaking forward, the absolute dipole values are usually ordered as Q1 (Al) < Q1 (P) < Q1 (O) [13] (see average
dipole Q1 notation below). The distortions lead to a
more pronounced dipole on P in the smaller PO4 unit
than on the larger Al one. So, a better EP representation using only lowest moments (L = 1) is possible
mainly due to the O contribution. This partly justifies
that only O atoms were often intuitively used (together
with cations if the zeolite or sieve has Al/Si and Al/P
ratios different from unity) by researchers in computations within empirical and semi-empirical pair-wise
addition schemes of the electric field influencing any
adsorbed particle or molecule.
In order to derive a simple representation for the
atomic dipole of zeolite oxygens, we first choose to
determine its absolute value Q1 = ((Q01 )2 + (Q11 )2 +
2 1/2 , where Qm is the m-component of the
(Q−1
L
1 ) )
L-order atomic multipole moment in a.u. [6,9]. We
propose that the dipole of the O atom within a chemically bonded moiety Al–O–P should be proportional
to the total dipole moment of this system represented
crudely as an “isolated” one. The total dipole should
be proportional to the bond anisotropy 1R which leads
to a non-zero dipole value even for the bond angle
ϑ = π. This means that terms proportional to ϑ
and to 1R should be considered separately in a first
approximation. This simple idea allows us to propose a five-parameter expression with respect to two
variables only (1R, ϑ) for all 106 atomic dipole values (not considering AlPO4 -34) calculated with the
STO-3G basis:
Q1 (1R, ϑ) = a1 en(1R−R0 ) + a2 sin(ϑ − ϑ0 )
(3)
where a1 = 2.075, n = 1.0616, R0 = 2.656 Å, a2 =
0.048, and ϑ0 = −0.0282 rad. Such approach provided a satisfactory RMSD value = 2.10%. A poorer
coincidence was obtained with function (3) when considering rather long or too short average bond R values. The negative deviations for the longer average
bond lengths and the positive deviations for the shorter
ones were avoided by introducing an additional simple power term function of a third variable R:
Q1 (R, 1R, ϑ) = a1 en(1R−R0 ) + a2 sin(ϑ − ϑ0 )
+a3 (R − R00 )m
(4)
Fitting with Eq. (4) within the STO-3G basis resulted in RMSD = 1.67% and parameter values
a1 = 1.272, n = 0.973, R0 = 2.30 Å, a2 = 0.0467,
ϑ0 = −0.222 rad, a3 = −0.3851, R00 = 1.430 Å, and
m = 3.038. Any exponential function as a third term
in Eq. (4) did not lead to a better result.
The absolute dipole values obtained within both
split-valence bases could also be satisfactorily
approximated using form (4). However, it seems to be
excessive for such an approximation. The absolute values of the dipole moments calculated with the ps-21G∗
and 6-21G∗ basis sets decrease almost linearly with
angle ϑ (Fig. 4 and Table 5). This proves an unsaturation of the STO-3G basis to represent O dipole values
for higher bond angles. Despite a wide variation of the
1R variable (Table 5), only terms proportional to the
average distance R (but also small contribution) and
angle ϑ in the right-hand side of Eq. (4) remains valid
while fitting the dipole values. Respective parameters
are shown in Table 6. The comparison between the m
values confirms the softer geometry dependence with
R of the O atomic dipole obtained with ps-21G∗ or
6-21G∗ versus those obtained with STO-3G. This is
in line with the behaviour of the charge dependence
via Eq. (2). However, considering the very wide
variation of the dipole value (nearly ±70% around
average value for both the series obtained with the
two split-valence sets, Fig. 4), a simple function
Q1 (ϑ) = a sin ϑ + b
(5)
provides a precise enough estimation for the O dipole
moments with RMSD of 8.4% (a = 0.2112, b =
0.0263) with ps-21G∗ and 6.2% (a = 0.1385, b =
0.0255) with 6-21G∗ . The reasons of such a difference
between the computed moments and those approximated by means of formula (5) come from different
points. For the dipole series computed with 6-21G∗ ,
the deviations of −16 and 13% comes from O(3) of
AST and O(2) of ATO with moderate angles of 146.4
and 162.9◦ , respectively. In ps-21G∗ , two serious deflections of 11 and 25% correspond to small dipole
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
83
Fig. 4. Absolute dipole moments calculated with the STO-3G (filled squares), ps-21G∗ (open squares), and 6-21G∗ (circles) basis sets
with respect to Al–O–P angle. Linear approximations are given by dotted, dashed, and solid lines, respectively. Oxygen dipole values for
the optimised CHA structure are marked for the 6-21G∗ basis.
for ϑ near 180◦ for O(2) and O(3) of AST, as already
discussed for the atomic charge in Section 3 (Table 4).
The latter present analogous internal geometries but
opposite AlO4 and PO4 tetrahedral symmetries (C3v
or Td ). We hope that a final explanation of the deviations of the approximated O(2) and O(3) dipoles
versus the calculated ones could be proposed together
with a closer analysis of the Al and P dipole moments,
which is presently under study [13]. A priori, the
high differences with the fitted dipole values could be
qualitatively explained through a strict Td symmetry
of the neighbour TO4 tetrahedra. In order to demon-
Table 5
Absolute dipole moment values Q1 of four ALPO sieves calculated with STO-3G, ps-21G∗ and 6-21G∗ basis sets with respect to the
average bond length R, bond anisotropy 1R, and Al–O–P angle
Type
R (Å)
1R (Å)
ϑ (rad)
ps-21G∗
6-21G∗
STO-3G
AST
1.5595
1.5618
1.6119
0.2960
0.3117
0.3563
3.1416
3.1413
2.5548
0.02958
0.03552
0.13630
0.02473
0.02662
0.08835
0.17553
0.18377
0.20487
ATN
1.5788
1.5900
1.6213
1.6227
0.2424
0.1988
0.2238
0.2196
2.6987
2.6332
2.7150
2.5920
0.10843
0.12247
0.10843
0.13335
0.08408
0.09109
0.07886
0.09596
0.17883
0.17494
0.17545
0.18231
ATO
1.6284
1.6361
1.6484
1.6636
0.1348
0.1271
0.1392
0.1459
2.5712
2.4468
2.8431
2.2967
0.13531
0.17077
0.08493
0.19746
0.10394
0.12034
0.07613
0.13601
0.16963
0.17553
0.15369
0.18231
AlPO4 -34
1.6262
1.6281
1.6401
1.6425
0.2000
0.2007
0.2086
0.2083
2.5789
2.5377
2.6377
2.5751
0.14341
0.15183
0.12599
0.13591
0.09916
0.10286
0.09212
0.09902
0.17950
0.18119
0.17512
0.17904
84
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
Table 6
Parameters of approximate function (Eq. (4) without the first term
proportional to 1R) evaluated using 15 crystallographically independent oxygen dipoles of ALPO sieves with the ps-21G∗ and
6-21G∗ basis sets
Basis set a2
ϑ 0 (rad) a3
ps-21G∗ −0.2235 2.224
6-21G∗ −0.1311 2.159
R00 (Å) m
0.2313 1.559
0.1644 1.488
sentation as in Fig. 3d. Similar type of analytical expressions for all components of the dipole and upper
moments up to octupole for O, Al, and P atoms are
possible. This work is actually under progress.
RMSD (%)
0.0374 3.9
0.0293 5.8
strate this, let us again consider an “isolated” fragment
Al–O–P whose total dipole moment is related to the
oxygen. An expression for any central moment QL
(which we can assign to the oxygen) of this moiety
through local moments Ql of lower or equal order
l ≤ L was developed by Stone (formula (11) in [25]).
Zeroth values of all multipole moments QL , L < 3,
on the T atom within TO4 tetrahedra of Td symmetry
[26] lead to a smaller total multipole of the same order
on the central O atom. Within AST, the contributions
to the central dipole moment will be zero from the
local P(2) dipole moment for the O(2) dipole, or from
the local Al(2) dipole moment for O(3). The difference between the atomic dipole values of Al and P is
the reason of the contrary signs of the dipole deflections obtained with approximate function (4). That
is why this simple analysis requires a more detailed
explanation including Al and P representations.
Another useful point to remark is related to the
comparison of the optimised (CHA from [19]) and
non-optimised (AST, ATN, ATO) models for a common search of the dependences. Only the CHA model
has been optimised, that is why all models are used
together with results of X-ray experiments considered
without further optimisation. The behaviour versus the
Al–O–P angle of the respective four O dipole values of
CHA, ranging between 145.53 and 151.16 (2.54 and
2.64 rad in Fig. 4), is similar to the one of the other
structures (we did not mark them by separate symbols
for simplification in Fig. 4). Evidently, the parameters
of the approximate dependences should be more reliable than those obtained after fitting of the moments
related to “raw” (non-optimised) zeolite models, but it
should not lead to any serious quantitative difference.
Ideally, the approximation for the estimation of the
O dipole moment shown herein should be complemented by an analogous representation of the upper
atomic moments in order to obtain a more precise pre-
5. Conclusion
A distributed multipole analysis on the basis of periodic Hartree–Fock calculations with three levels of
basis sets, the minimal STO-3G, a split-valence including pseudo-potential (Durant–Barthelat) ps-21G∗
basis on aluminium and phosphorus and 6-21G∗ on
oxygen, and the split-valence 6-21G∗ on all atoms,
via the CRYSTAL95 code was considered for 12 aluminophosphate (ALPO) molecular sieves with ratio
Al/P = 1. With the extended basis set, convergence
of the SCF procedure could be reached for the sieves
AST, ATN, ATO, and AlPO4 -34 only. 110 and 15 Mulliken atomic O charges for the crystallographically independent O types were computed with the STO-3G
and ps-21G∗ or 6-21G∗ basis sets, respectively. Their
analysis were then performed in terms of three internal parameters, the average T–O distance (T = Al, P),
the T–O bond anisotropy, and the Al–O–P bond angle
of the framework oxygens. The best fitting of the O
charges is obtained via a six-parameter approximate
expression (Eq. (2)).
An analogous five-parameter approximate function
(Eq. (4) without first term in the right-hand side) for
the absolute dipole moment value of the O atoms was
obtained first with respect to the same three internal
coordinates as for the Mulliken O charges. A higher
importance of the term corresponding to the angular
dependence of the dipole was, however, observed with
the split-valence bases as compared to the STO-3G
one. Several deviations of the approximate function
values for both the O charges and dipole moments
here calculated were observed for different symmetries of the two closest neighbour TO4 tetrahedra. Further studies of the relation between the TO4 symmetry
variation and charge values would be useful to derive
more precise expressions.
The fitting with function (4) may be considered as
a first satisfactory approximation of the multipole moments of orders higher than zero (i.e. of the charges)
for all O framework atoms. It suggests that approximate functions could also be fitted for other higher
A.V. Larin, D.P. Vercauteren / Journal of Molecular Catalysis A: Chemical 166 (2001) 73–85
moments assuming a deeper theoretical approach to
search the respective functions. Then a precise representation of the electrostatic field within frameworks
with a larger number of atoms per unit cell would be
possible even if one cannot calculate the respective
field directly with codes such as CRYSTAL.
Both the approximate forms for the charges and
dipole moment could be useful for the field estimations
in any ALPO framework while constructing embedded
model or using empirical simulation of any ALPO
sieve.
Acknowledgements
The authors wish to thank the FUNDP for the
use of the Namur Scientific Computing Facility
(SCF) Centre, a common project between the FNRS,
IBM-Belgium, and FUNDP. They acknowledge financial support of the FNRS-FRFC, the “Loterie
Nationale” for the convention no. 9.4595.96, and
MSI for the use of their data in the framework of the
“Catalysis and Sorption” consortium. AVL acknowledges the PAI 4-10, the “Services du Premier Ministre
des Affaires Scientifiques, Techniques et Culturelles”
of Belgium for his Postdoctoral stage and the Russian
Foundation of Basic Researches for financial support
(Grant no. 96-03-33771).
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